Consider the heat equation \(u_t-\Delta u=0\) on a bounded \(C^2\) domain \(\Omega \) in \({\mathbb {R}}^{n}(n\ge 2)\) with any positive initial data. If a superlinear radiation law \(\frac{\partial u}{\partial n}=u^{q}\) with \(q>1\) is imposed on a partial boundary \(\Gamma _1\subseteq \partial \Omega \) which has a positive surface area, then it has been known that the solution u blows up in finite time. However, if the partial boundary, on which the superlinear radiation law is prescribed, is shrinking and is denoted as \(\Gamma _{1,t}\) at time t, then the solution may exist globally as long as the surface area \(|\Gamma _{1,t}|\) of \(\Gamma _{1,t}\) decays fast enough. By taking advantage of the Neumann heat kernel, we conclude that a polynomial decay \(|\Gamma _{1,t}|\sim |\Gamma _1|(1+Ct)^{-\beta }\) with any \(\beta >n-1\) suffices to ensure a bounded global solution.
